I'm highlighting this claim about Fixed Odds Betting Terminals in particular because it is so mathematically wrong-headed. Also because the "gambling away £100 every 20 seconds" and "gambling away £18,000 per hour" narratives are now forming a core part of the campaign against FOBTs. I may come back to the political and moral arguments on this subject in a later post. But for now I just want to focus on the maths.

FOBTs, which are most commonly roulette have a house edge of between 2.7% and around 5%. This is in line with the standard odds yielded by a roulette table. If a table has a single zero then the odds of winning on a red or black bet are just less than 50/50 (because if it comes up zero the house wins). So with 36 numbers plus zero the chances of winning in a single spin are 18/37 or 0.486. This yields a house edge of just over 2.7%. If the roulette table has two zeroes (a zero and a double zero) then the odds are titled more in the house's favour as it becomes 18/38 or 0.474 yielding a house edge of 5.26%.

So as probability theory tells us, over the long term we would expect a punter on the first type of table to be about 2.7% down and a punter on the second type of table to be 5.26% down. For the rest of this piece I am going to focus on the second type of table as I wanted to pick the worst case scenario for this to be generous.

If someone was foolhardy enough to bet £100 on a red or black bet every 20 seconds for an hour on this sort of table they would be incredibly unlikely to lose (or "gamble away" to use Watson's term) £18,000. In fact we can calculate just how unlikely this is. We simply take the chance of them losing each time and multiply this number by itself the number of times they are going to play. From this we can see that the chance of this happening is (1-(18/38))^180

Which is one in 149,847,041,024,310,787,847,729,246,045,620,000,000,000,000,000,000*

Or to put it another way roughly the same likelihood as picking two random molecules of water from all the oceans of the world and them being the same one. In other words pretty much as close to impossible as you can theoretically get.

In actual fact someone who gambled £100 every 20 seconds for an hour would not have "gambled away £18,000". They would, on average gamble away 5.26% of £18,000 or £946.80. Now I'm not trying to claim this is not a large amount of money, it of course is, but it's only around a twentieth of the amount that the headlines and distortions of the campaign would have you believe.

More importantly though it is the most extreme interpretation of the possibilities. How many people who gamble on these machines seriously wager the maximum amount every 20 seconds for an hour (or more) at a time? I suspect very few. Far more likely is that they would wager smaller amounts, perhaps £5 or £10 a spin. So if someone did bet £5 for 180 spins they'd be on average down about £48. Now whether £48 spent in this way for an hour is a wise use of money is a different question. But of course losing £48 in an hour is not as scary or headline grabby as £18,000 in an hour.

As I said the politics and morality is for another post and I am not covering that here. I'm simply saying that if people want to campaign on this they should at least get the basic maths right.

*After the 32nd figure my calculator ran out of accuracy (because the number is so massive) so the final 19 digits are all zeroes meaning that it is rounded down.

This is all absolutely correct and the figure quoted (not just by Watson, I see, is obviously misleading).

I think however this too might be misleading

Play for long enough and most of the time you will more-or-less break even. Betting might still be a mug’s game but FOBTs offer you a better chance than the horses. (Spending £1000 on a FOBT can, all things being equal, be expected to cost you £30. The average player, of course, will spend much less than that on their gaming. And so lose less.) - A.Massie, Spectator

In essence what he's saying here is if you want to bet £100, there's a 97% payout ratio so not much to worry about. But isn't the problem that because you can repeatedly gamble that money, you might actually end up having gambled far more than your original stake (which might not matter) but losing almost all or your original stake (which might)

Of course it still means the notional amount quoted by Watson is idiotic, but it also means people who don't have much money can gamble very high notional amounts, which might make their actual loss much higher than 97% implies.

@Matthew: no it doesn't make sense. If you don't have very much money to gamble thats all you can lose, whatever the theoretically possible amount you 'could' gamble in any given time.

If you gamble in the way stated - make multiple bets over a long period of time, the law of averages determine that you can only lose a relatively small % of the cash you started with. It doesn't make any difference if its £100 in £1 bets, £500 in £5 bets or £1000 in £10 bets. Your loss is always limited to the amount of cash you start with, and if you limit your wagers to a small % of your initial pot, you will be able to gamble for a fairly long period of time and lose less than your initial cash pile.

Jim, I don't quite follow. I know you can't lose more than your original pot, but what I'm wondering is whether that really equates to 'most of the time you will more or less break even'. I think that's true if you measure your loss as a % of the notional gambled, but not if you measure it as a % of the original pot.

It depends on the type of gambling of course. In this case one is told one can only gamble £100 per 20 seconds on low odds games, such as red or black, or odds or evens. The odds are therefore ever so slightly below 50/50. Ergo if you gamble any stake for a long period, the law of averages kicks and and you will win some, just slightly less than you lose. Thus if I bet £100 every time on red or black, by the time I've done 10 bets its highly unlikely I will have lost every one of them. I may be down but not down by £1000. Do that 100 times and the actual returns get closer and closer to the theoretical ones. As the house's edge on average kicks in once in 37, or roughly every 12 minutes, and you lose your stake when that happens, you would expect to lose a % of your pot to the house that often. The exact % being however much your stake is as a percentage of your initial pot.

For example if you start with £100 and bet £2 every time, after 12 mins theoretically you should have won 18 times, lost 18 times and once lost your stake to the house, ie have £98 quid left. Of course that is in a perfect world - most will be above or below that figure. However the point is that all will be above £26 left, which is what the article would assume - 37 spins, £2 each time = £74 spent. They ignore all the winning bets.

Good points Mark about the stats. Just 2 thoughts :1. Aren't all stats used by politicians bollocks?2. There is also the chance that the person betting could be a "big winner" but only if "you're lucky!"

I think the problem is that people are not backing low risk outside bets; they're laying high risk inside bets backed by relatively limited bankrolls. (Hence the problem in these machines having become so prevalent in areas where pockets aren't deep and statistical understanding is not a quality of which most people are possessed...) Article is spot on in dismissing Watson's claims (although one could be generous and say 'gambling away' does not necessarily mean 'losing') However, the reality is that a 100% loss every 20 seconds is, by some way, the most common outcome for the type of bets that are actually being placed. Losing 18,000 an hour would indeed be extraordinary, but the speed of play means bankrolls are frequently broken extremely (and I mean unbelievably) quickly - even at low stakes; also, any wins that do occur can be reinvested around 4 times as quickly as they would be in a casino. If this isn't a recipe for problem gambling, I don't know what is. That being said, there is a simple remedy - make it easier for people to collect their money: bring spin times in line with casinos and offer all wins for collection. Sod it, remove the stake cap altogether as a way of placating the bookies. At least then nobody can claim the player doesn't have the time, or the freedom, to make rational decisions.